This week has seen a dozen requests from radio stations and the press for comment about the same thing: what are we going to call the new decade? There was a similar fuss a decade ago, I recall, when the noughts (noughties, etc) were being debated. An Australian initative a few weeks ago asked for popular suggestions. The winner was one-ders - a piece of word play involving wonders and the ones which will be part of every year. Other suggestions were decnos, tentions, tweens, tennies, and twenteens.
I don't think human linguistic nature has changed much in the past century, so my guess is that what will happen today is the same as happened then. During the middle years of the century, people talked a lot about 'the tens, twenties, and thirties'. Tens was the predominant usage. However, there was also quite a lot of reference to the teens - the OED has citations dating from the 1930s. So the choice, it seems to me, will be between those two. If I had to choose, I would bet on tens, because these days teens has the dominant sense of teenagers, and people may well avoid using it for that reason. But it's not wise to bet, where language change is concerned.
Of course, this is all to do with informal usage. At a formal level, the issue is clearer. We have the choice of two thousand and ten (in British English - two thousand ten in American English) and twenty ten. Again, based on past centuries, speech is more likely to go for the shorter version. It's rare to hear 'in nineteen hundred and ten'. And I've never heard Tchaikovsky's overture called 'eighteen hundred and twelve'.