As David pointed out, you basically ask for bounds on the treewidth of a connected graph with average degree 3. For the more special case of 3-regular graphs, the following lower and upper bounds can be obtained. Denoting by pw(G) the pathwidth of a graph G, it is clear that
(1) tw(G) <= pw(G) for any graph G (as a path decomposition is a tree decomposition)
It is proved in  that
(2) For every \epsilon > 0, there exists an integer n_0 such that for any 3-regular graph G on n >= n_0 vertices, pw(G) <= n/6 + \epsilon*n.
This gives you an upper bound of roughly n/6 on the treewidth of 3-regular graphs.
For an almost sure lower bound, I cite from :
"As random cubic graphs almost surely have bisection width at least 0.101 n (Kostochka, Melnikov, 1992), they have almost surely no separator of size smaller than n/20"
and thus almost surely no tree decomposition of width smaller than n/20.
For a "sure" lower bound on the bisection width,  showed an infinite family of 3-regular graphs, such that each graph G=(V,E) in this family has bisection width at least 0.082 * |V|.
 Fedor V. Fomin, Kjartan Høie: Pathwidth of cubic graphs and exact algorithms. Inf. Process. Lett. 97(5): 191-196 (2006)
 Jaroslav Nesetril, Patrice Ossona de Mendez: Grad and classes with bounded expansion II. Algorithmic aspects. Eur. J. Comb. 29(3): 777-791 (2008)
 Sergei L. Bezrukov, Robert Elsässer, Burkhard Monien, Robert Preis, Jean-Pierre Tillich: New spectral lower bounds on the bisection width of graphs. Theor. Comput. Sci. 320(2-3): 155-174 (2004)