There's a famous rule which states that any 2d maze can be solved by utilizing an "always go right" decision making strategy and "reverse once you reach a dead end" decision making strategy.
I believe it is impossible to create a rule of this "spirit" in 3 dimensions but i'm trying to turn that into a more rigorous statement and proof.
Here's what I got:
Ultimately at any intersection point in the maze the "simple rule" is to create set of relative coordinates, and pick a path according to those coordinates using a relative ordering of paths w.r.t to the coordinates.
In the 2D case whenever we hit a junction of multiple paths, we can view the intersection point as having an action of a cyclic group generated by the "pick the closest path to the right" generating element. Thus this generator "pick the closest path to the right" allows us to solve 2D mazes.
In the 3D case [this is where I fail to clearly state what I want to say] the space of paths doesn't have a natural action of a cyclic symmetry group on it, so its not possible to find a simple rule to generate all the options, and therefore there is no simple "right hand rule"-analog for solving the 3 Dimensional case.
Though I don't have any insight into "action of any group" on paths while on junction ...
The 2-D rule obviously can't work, and it does not. Imagine the right-loop exiting from the junction.
The "always go right" in my view is just ordering of the possible exits from the junction. You then traverse the picked path to the next junction and so on. Sometimes you come to a dead end or to an already visited junction, then you backtrack from that location and pick the next exit at the last junction, while marking the checked path as not taking you to the solution.
The same strategy will work in any dimension, being it $3$ or higher. You can call it a "brute-force, enumerate all possible paths" solution. If there is a path to goal you will find it eventually, or you will go over all possible paths and understand the goal is unreachable.
UPDATE [2023-07-16]: As of now, we have that no automaton can solve any 2D maze [Budach, 1978], however those that can traverse, can be reduced to ones that follow "left-" or "right-hand-on-the-wall" rule [Kilibarda, 2017], as you hint. In the general graph settings, you either pre-label the exits [Gąsieniec, 2007, Ilcinkas, 2008], which you basically suggest, or you need a memory that depends on the graph characteristics [Fraigniaud, 2005], e.g. diameter, largest in- or out-degree, number of vertices, etc.
However, if you just assume, that you are in the sub-graph of some 3D cube grid graph, then oblivious travel inside is extremely limited [Missing reference, but visiting all nodes is possible in simple cubes which are of maximum [ordered] size of $n \times 3 \times 3$]. This is not exactly what you have asked, but could be very close.
Budach, Lothar, Automata and labyrinths, Math. Nachr. 86, 195-282 (1978). ZBL0405.68049.
Fraigniaud, Pierre; Ilcinkas, David; Rajsbaum, Sergio; Tixeuil, Sébastien, Space lower bounds for graph exploration via reduced automata, Pelc, Andrzej (ed.) et al., Structural information and communication complexity. 12th international colloquium, SIROCCO 2005, Mont Saint-Michel, France, May 24–26, 2005. Proceedings. Berlin: Springer (ISBN 3-540-26052-8/pbk). Lecture Notes in Computer Science 3499, 140-154 (2005). ZBL1085.68110.
Gąsieniec, Leszek; Klasing, Ralf; Martin, Russell; Navarra, Alfredo; Zhang, Xiaohui, Fast periodic graph exploration with constant memory, Prencipe, Giuseppe (ed.) et al., Structural information and communication complexity. 14th international colloquium, SIROCCO 2007, Castiglioncello, Italy, June 5–8, 2007. Proceedings. Berlin: Springer (ISBN 978-3-540-72918-1/pbk). Lecture Notes in Computer Science 4474, 26-40 (2007). ZBL1201.68147.
Ilcinkas, David, Setting port numbers for fast graph exploration, Theor. Comput. Sci. 401, No. 1-3, 236-242 (2008). ZBL1147.68042.
Kilibarda, Goran, On reduction of automata in labyrinths, Publ. Inst. Math., Nouv. Sér. 101(115), 47-63 (2017). ZBL1474.68173.