We give an elementary new method for obtaining rigorous lower bounds on the
connective constant for self-avoiding walks on the hypercubic lattice Z a. The
method is based on loop erasure and restoration, and does not require exact
enumeration data. Our bounds are best for high d, and in fact agree with the
first four terms of the lid expansion for the connective constant. The bounds are
the best to date for dimensions d>~ 3, but do not produce good results in two
dimensions. For d= 3, 4, 5, and 6, respectively, our lower bound is within 2.4%,
0.43%, 0.12%, and 0.044% of the value estimated by series extrapolation.