Is this a case for ordered logistic regression? I have the following study setup:
Three groups of people were asked a question, and the answer was ordinal (likely, somewhat likely, somewhat unlikely, unlikely). In my data set, I have a contingency table between the response and groups, and would like to know whether some of the groups of people were more inclined to answer the question with likely than others.
EDIT: I didn't clearly state this above, but I want to treat the answer as ordinal, i.e. likely = 1 < somewhat likely < somewhat unlikely < unlikely.
I think I could do it with an ordered logistic regression model, but isn't that like shooting cannon at sparrows?
 A: I think an (ordered) logistic model is certainly warranted. You could calculate a Chi-square for the table first, to get a feeling whether there is any connection at all. You could also have a go at calculating odds ratios for the table, but this is in principle the same as the regression. 
A: 
would like to know whether some of the groups of people were more inclined to answer the question with likely than others

Well, as you anticipate, you can answer that specific question without going all the way to ordinal logistic regression.
If you regard your table "answered 'likely'" vs "answered one of the less-likely categories" (i.e. collapse the lower categories), your direct question is a simple chi-square - any differences in the proportions answering "likely" among the three groups show up as a difference in the chi-square test for independence.
(Further, two-sample proportions tests could be used as a basis for post-hoc tests of differences in similar fashion to ordinary ANOVA.)
That's not to dismiss ordinal logistic regression - but it's not required for that particular question, and a simpler analysis has the benefit that it's likely to be well understood by a larger audience.

I considered this route, but I would rather look for a general tendency towards likely (so, "somewhat likely" counts more than "somewhat unlikely").

In this case your response is ordinal, and you're testing for a general location difference. While this could be done with ordinal logistic regression, again, you don't need to do that; simpler machinery can suffice (and be more readily interpretable).
You could, for example consider rank-based tests, which work on ordinal data. Your only issue is that you're dealing with heavy ties. Nevertheless it's perfectly possible to do permutation/randomization tests using the statistic. In this case, you could for example compute a Wilcoxon-Mann-Whitney type statistic (well, with three groups, a Kruskal-Wallis type statistic), and then get the null distribution by randomly reallocating group labels and computing the statistic again (repeatedly) for a randomization test. (Or if the number of observations is small, you can calculate all possible reallocations of the group labels.)
That would yield an exact nonparametric test of location, with perfect adjustment for ties.
