Since I wrote about an artificial 2C response to 1-of-a-major, some people have asked me about continuations. There are certainly plenty of options. There are a few complete schemes available on the web, but I'd imagine that for most people who want to play this sort of thing, part of the fun of the method is in designing their own structure - and doing it yourself also makes it easier to remember (though perhaps not for partner!) Still, there are a few things that can be said about how best to go about it.
First of all, natural bidding works adequately well and is a good place to start. The important thing, in my opinion, is to distinguish responder's balanced hands from hands with clubs as soon as possible, and if you play natural continuations this means responder always bidding 2NT at his next turn if this is available. It follows that you can have a problem if opener's reply to the 2C bid is 2NT or higher - these rebids do not allow an easy, natural way to continue. So ideally these replies should not be too frequent, and the natural meaning probably is a little too frequent.
At the same time, opener's 2D rebid is rather underused if played as natural. So a big improvement on strictly natural methods is to bundle some more hand types into 2D. Glen Ashton has written up a convention he calls "2Dlay" (see here) where the 2D bid shows a hand which would have rebid 2D or 2NT or 3C playing natural methods. After the 2D bid, responder can rebid 2H to ask which hand type is held (with 2S showing most hands with diamonds). If you are looking for a simple approach, this is a very good idea, since it makes the system much more efficient while still quickly leading to natural bidding later in the auction.
However, natural-based continuations have their faults. My main concern is that there is often no easy way for either partner to show the strength of their hand. This is a common problem in 2/1-based systems: hands with extra values can be difficult to bid. I feel it is much better to show something about strength explicitly as soon as possible, and this can only be done using artificial methods. Now, you can arrange things so that opener describes his strength, or so that responder describes his strength. I prefer it to be opener, because responder has some catching up to do in terms of describing his shape, and having to have two ways of showing a balanced hand would make life difficult in various ways.
But since showing shape is also very important, the description of strength cannot be too detailed, and so nearly all the methods I have seen divide opener's strength into just two ranges. For simplicity we can call the ranges "minimum" and "maximum", though if the opening bid is wide-ranging this is a bit misleading - the upper range would typically start at about 14 HCP, and the very strongest hands would have to make a further move later.
Playing artificial relay-like continuations, there is actually enough space available for opener to describe his complete shape, as well as whether he is minimum or maximum, below game level. This is what responder would want to happen whenever he has a very strong balanced hand. However, being able to do this is not the only important consideration when devising continuations. There are two reasons why responder may not want opener to describe his hand completely: firstly, it may be possible to name the final contract without having had a complete description (and further information would only be helpful to the opponents), and secondly, when responder holds an unbalanced hand he might want to make a descriptive bid himself.
Trying to take these things into account, let's look at a method which is based around the following rebids for opener:
2D = any minimum
2H = maximum, balanced or 4+ cards in a minor
2S = maximum, 4+ cards in the other major
2NT = maximum, 6+ cards in the suit opened, not 4+ in the other major
After 2H or 2S, 2NT will be a further asking bid (implying a balanced hand), whereas after 2NT, balanced hands will have to continue by bidding 3C. More interesting is the scheme after 2D: using 2H as responder's next relay, opener will reply:
2S = balanced or 4+ cards in a minor
2NT = 6+ cards in the suit opened, not 4+ in the other major
3C+ = 4+ cards in the other major: bids show the same shapes as after 1M : 2C , 2S : 2NT.
Notice the symmetry here: once opener has begun to show shape, any further bids are the same for minimum hands as for maximums.
This scheme works particularly well when it comes to responder breaking the chain of relays. As said above, the first reason responder might want to do this is if he already has enough information to be able to name the final contract. In order to achieve this, the replies to 2C are arranged so that we get the most important information first. In particular, it is very useful to know immediately if opener is minimum, since responder is unlikely to be able to sign off confidently if he does not have that information. A common auction is 1M : 2C , 2D : 2H , 2S, where opener has shown a minimum and denied four cards in the other major. This may well be enough for responder to place the contract in 3NT or 4M. [Actually, making one more relay is more common, since opener could still have an extreme shape such as 6-5 with a 5-card minor. One useful idea is to use 3D as a "weak relay" in this auction, asking opener whether he has a 5-5 shape, promising that responder will be able to set the contract otherwise. This gives away the minimum amount of information.]
It is also important for opener to make a good start at describing his hand in case responder is unbalanced. If this happens then relays will stop, and the partnership will revert to natural-based bidding. So we want to ensure that opener's first reply to 2C does not make subsequent natural bidding too difficult. Most importantly, opener's more space-consuming replies must be very well defined. This is why, in the scheme above, 2S shows a more specific hand type than 2H. Over 2H, responder can bid 2S with an unbalanced hand (artificial showing 5+ clubs), which leaves room for opener's hand type to be revealed. This would not be possible if 2H and 2S were reversed.
Some aspects of opener's hand are particularly difficult to describe using natural bidding, and so we need to use the reply to 2C to help with this, in case responder declines to relay afterwards. This is the main reason why the very first thing we do is distinguish between minimum and maximum hands: showing strength is very difficult in natural bidding, particularly if you are unable to identify a trump suit quickly. Another thing which is difficult to show naturally is a hand of 5-5 shape. In order to describe these fully in natural methods, you would have to bid and rebid the second suit. So ideally, when you hold a 5-5 hand you would want your reply to 2C to show the second suit. However, in the scheme above, we only do this on maximum hands with both majors. Other two-suiters can cause a problem if responder needs to know about the fifth card in the second suit. This is particularly likely to be problematic if the second suit is the other major. For this reason, it seems to be a good idea to use opener's currently undefined 3C response to show a minimum hand with at least 5-5 in the majors (5-6 after a 1H opening).
When working out the replies to responder's 2NT or 3C relays, I feel that showing shortage is most useful. So for example, after 1M : 2C , 2H : 2NT (and 1M : 2C , 2D : 2H , 2S : 2NT) we could use
3C = no shortage (i.e. 5-3-3-2, or 5-4-2-2 with a 4-card minor)
3D = 5+ diamonds
3H = shortage in the other major
3S = shortage in diamonds
3NT = shortage in clubs
[This assumes that 5-5 hands with clubs are put somewhere else: this is possible if you use 1M : 2C , 3D for maximums and 1M : 2C , 2D : 2H , 3D for minimums. The latter sequence is not needed for a major two-suiter if that hand would bid 3C directly over 2C.]
Further asking bids are possible over 3C, 3D and 3H, but usually once opener has shown shortage it should be possible for responder to work out what the best game should be, and in particular whether 3NT will be a good contract.
After that you would need some slam-bidding methods. The sort of slam-bidding conventions you find in natural systems aren't really appropriate here. You can go a long way just using 4C as asking about general strength. I'll write a post about this at some point.