We say semimonthly when we describe something happening twice a month. for example "a semimonthly publication", "a semimonthly meeting", etc. Is there any word we can use to describe something that happens three times a month; preferably one word?
Trimonthly is, in fact, already in several dictionaries. Problem is, unlike its counterparts bimonthly and biweekly, trimonthly is only defined as occurring every three months, sans a secondary three-times-a-month meaning.
I would recommend thrice-monthly, if you needed to form a single word.
The committee meetings will occur thrice-monthly until this matter is resolved.
As was pointed out in the comment below, the hyphen could be omitted as well.
Also, I'll restate my opening remarks, but add some links:
Although bimonthly can be used either way (see here), that doesn't appear to be the case for trimonthly (see here, here, here, and here). In light of such evidence, I'd be hesitant to use trimonthly when meaning "three times each month."
You might consider the rare trimensual.
These forms also exist for the month, although rarely used in English. Nevertheless, I think they sound right, so to speak.
bimensual (adj.), bimensually (adv.) = twice a month
bimestral (adj.), bimestrally (adv.) = every other month
trimensual (adj.), trimensually (adv.) = three times a month
trimestral (adj.), trimestrally (adv.) = every third month
(Note the connection to the word trimester.) Latin had no week, and it seems especially silly to slice up a day in this way (as opposed to speaking of hours), so I guess that's pretty much as far as it goes.