We demonstrate tunable spin-spin couplings between trapped atomic ions, mediated by laser forces on multiple transverse collective modes of motion. A sigma(x)sigma(x)-type Ising interaction is realized between quantum bits stored in the ground hyperfine clock states of Yb-171(+) ions. We demonstrate entangling gates and tailor the spin-spin couplings with two and three trapped ions. The use of closely spaced transverse modes provides a new class of interactions relevant to quantum computing and simulation with large collections of ions in a single crystal.