It was 3-D World Hunt. See this archived NYT page, reproduced in part below.
December 17, 2006
3-D Word Hunt
By Will Shortz
Imagine that the lettered balls in the three-dimensional grid at right are arranged in two vertical layers: U, D, I, R, A, L, P, H and E in front and M, E, N, O, T, S, A, C and W in back. Find as many words of exactly five letters as you can spell by following the rods connecting the balls. You may return to a letter and use it twice in one word (as the T in STATE), but you may not “stand” on a letter and use it twice before proceeding (as the T in LATTE). Plurals are fine. Proper names and prefixes are prohibited. Finding 25 words is par; 35 is expert; 45 is genius-level. Our answer list, containing 51 relatively familiar words and five not-so-common ones — all from Merriam-Webster’s Collegiate Dictionary, 11th edition — will be shown next week.