String theory uses a Feynman diagram with surfaces instead of lines, and world sheets instead of world lines. Can there be theories with solids instead of surfaces and world solids instead of world sheets?
The Standard Model(SM) of physics with its SU(3)xSU(2)xU(1) group structure is very successful in unifying the three forces in the particle domain, and string theories have the theory "phase-space" to embed the SM and also a quantized gravitation because it has niches for a spin 2 graviton. String theories are the only ones in the market that could offer a model for the Theory of Everything (TOF). A definitive model has not been picked up yet from the thousands of possibilities that can give predictions testable in the laboratory.
The SM "lives" in the four dimensional space of 3 space and one time dimension, and the elementary particles on which it is based are points.
String theories assume the elementary particles are vibrational levels on a string and go to high dimensions so that the group structure of the vibrations can embed the SM particles.
All string theory models need extra dimensions over the 3 space and 1 time, which at the level of phenomenology, i.e. predicting data, they compactify ( curl up)into volumes not separable from points, leaving only the 3 space and 1 time to model the world.
Taking strings, or membranes is for mathematical convenience in trying to correspond string vibrational spaces to the four dimensional world in which the SM data are measured and the models tested.
Introducing volumes ( which you must mean with your solids) or even higher than that dimensions instead of strings, would just have more dimensions compactified because the objective is to fit the data given by the SM, not to play with the mathematics.