The authors develop some efficient spectral algorithms based on Jacobi-Galerkin methods for the solution of integrated forms of fourth-order differential equations in one and two variables. The spatial approximation is based on Jacobi polynomials $P_n^{(\alpha ,\beta )}(x)$, with $\alpha ,\beta \in (-1,\infty )$ and $n$ the polynomial degree. For $\alpha =\beta $ , one recovers the ultraspherical polynomials (symmetric Jacobi polynomials) and for $\alpha =\beta =\mp \frac 12,\alpha =\beta =0$, the Chebyshev polynomials of the first and second kinds and Legendre polynomials, respectively. For the nonsymmetric Jacobi polynomials, the two important special cases $\alpha =-\beta =\pm \frac 12$ (Chebyshev polynomials of the third and fourth kinds) are also recovered. The two-dimensional version of the approximations is obtained by tensor products of the one-dimensional bases. The resulting discrete systems have specially structured matrices that can be efficiently inverted. An algebraic preconditioning yields a condition number of $O(N),$ ($N$ being the the number of retained modes of approximations) which is an improvement with respect to the well-known condition number $O(N^8)$ of spectral methods for biharmonic elliptic operators. The numerical complexity of the solver is proportional to $N^{d+1}$ for a $d$-dimensional problem. Numerical results are presented in which the usual exponential behaviour of spectral approximations is exhibited.