Gusty, try putting a fingertip lightly on the center of the head while you tap. This will give you more of a harmonic sound that may enable you to better hear the pitch. If that doesn't work, try muffling the head slightly with your fingers on the opposite side of the lug you're tuning, experiment with a little muffling or a lot to see what works better to hear the pitch.
Instead of a fingertip resting on the center of the drumhead, try a moongel. The pitch changes depending on how firmly you press your finger into the head, but the moongel remains constant.
As you tap, just listen for a pitch that is higher than the fundamental pitch of the drum. Zero in on it, and listen to that pitch at each lug. Just takes getting the hang of it.
Another mallet man. And I, too, don't reference a piano. Each drum has it's own "sweet spot", and I want the drum "to be all that it can be".
Yes. And once you find that sweet spot you can see what pitch it is, and thereby be able to easily replicate your tuning when it comes time to rehead. I do it with a pitch pipe.
Also, by identifying the pitch you've tuned to--and perhaps nudging it a little bit higher or lower while staying in the sweet spot--you can often arrange the spacing of the toms to be more even, more melodic, or whatever you want. For instance, a tritone between toms sounds pretty bad both sequentially and when the toms are played together; by nudging the tuning to a perfect fourth or a perfect fifth--whichever keeps the toms singing well--you get a much nicer result.
My kid's kit is 8-10-12-14f-16f, tuned to G, Eb, Bb, F, and C. All drums are in their sweet spots, yet you get a nice melodic sound around the toms, and any two adjacent toms sound great when struck together.
The key is to (as you say) let the drums tell you where they want to be tuned, but once that's done thinking in pitches is handy.