There seems to be no way of directly continuing the etymological pattern that formed 'dual' in English. Dual is derived from the latin dualis.
The equivalent number for for three in latin would be tres or tria. In English, we have the word 'trial'. We also have the prefix tri-, but that is not a word in and of itself.
In suspecting that trial would probably be the closest analog, I did come across 'paucal', which is listed as meaning:
- (grammar) pertaining to a language form referring to a few of something (three to around ten), as a small group of people; singular, dual, trial and plural.
3.(uncountable)(grammar) a language form referring to a few of something (three to around ten), as a small group of people; contrast singular, dual, trial and plural.
Excerpts from the Wikitionary entry of Paucal, licensed under the CC-SA-3.0 License.
It should be noted that I am not recommending 'paucal' as an alternative. It is a technical word with a restricted purpose that does not really match the desired definition, especially since it is less specific.
The reason I included the Wiktionary definitions of 'paucal', despite not recommending it, is that it suggests 'trial' would be the word to continue the pattern but again, if it were not for a couple of problems:
The linguistic word I found on Wiktionary suggests that a similar definition exists for trial, but that definition does not match either. I think this is actually more clearly demonstrated by the closest definition in the Randomhouse Kernerman Webster's College Dictionary, © 2010 definition of trial:
adj. 1. of or belonging to a grammatical category of number, as in some Papuan and Austronesian languages, used to indicate that a word denotes three persons or things.
More importantly, as I am sure you are all already well aware, 'trial' usually means something closer to "test" in English, and perhaps most especially the sort of test that requires adjudication in a court of law. I expect the word to be such a popularly and traditionally pervasive word that it is hard for me to endorse using the word to mean anything else, except perhaps by way of metaphor or metonymy.
As far as I can reliably discern, 'trial' does not even derive from tria, but the anglo-french triet which means something closer to try, as in an effort or attempt. Thus if you want a word which is in the same series as 'dual' that conveys the meaning you desire, I am currently of the opinion that a such word does not exist.
You may want to consider 'plural' as a possible alternative, but it is not as specific:
plural, adjective [Latin pluralis, from plus, pluris, more.]
- Containing more than one; consisting of two or more, or designating two or more; as a plural word.
The American Dictionary of the English Language by Noah Webster