We address the problem of efficiently testing for linear unboundedness and its applications to translational assembly planning. We describe a new algorithm that performs the test by solving a single homogeneous system of equations followed by a single linear feasibility test. We show that testing for unboundedness is computationally at least as hard as these two subproblems. The new algorithm is the fastest known algorithm and is practical. We then present a framework for general translational assembly planning based on linear constraints. We show the relation of m-handed assembly planning to unboundedness testing and present a polynomial-time algorithm for m-handed assembly of polygonal part assemblies with no initially separated pairs of parts. For the general translational assembly-planning problem, we present a new algorithm that uses unboundedness testing and a cell reduction technique to significantly increase the search efficiency. Experimental results of our implementation on a variety of planar and spatial assemblies demonstrate the practicality of the algorithms.