  The magnetic properties of a new family of molecular-based quasi-two
dimension $S=1/2$ Heisenberg antiferromagnets are reported. Three compounds,
($Cu(pz)_2(ClO_4)_2$, $Cu(pz)_2(BF_4)_2$, and $[Cu(pz)_2(NO_3)](PF_6)$) contain
similar planes of Cu$^{2+}$ ions linked into magnetically square lattices by
bridging pyrazine molecules (pz = $C_4H_4N_2$). The anions provide charge
balance as well as isolation between the layers. Single crystal measurements of
susceptibility and magnetization, as well as muon spin relaxation studies,
reveal low ratios of N\'{e}el temperatures to exchange strengths ($4.25 / 17.5
= 0.243$, $3.80/15.3=0.248$, and $3.05/10.8=0.282$, respectively) while the
ratio of the anisotropy fields $H_A$ (kOe) to the saturation field
$H_\mathrm{SAT}$ (kOe) are small ($2.6/490 = 5.3\times10^{-3}$,
$2.4/430=5.5\times10^{-3}$, and $0.07/300=2.3\times10^{-4}$, respectively),
demonstrating close approximations to a 2D Heisenberg model. The
susceptibilities of ClO$_4$ and BF$_4$ show evidence of an exchange anisotropy
crossover (Heisenberg to $XY$) at low temperatures; their ordering transitions
are primarily driven by the $XY$ behavior with the ultimate 3D transition
appearing parasitically. The PF$_6$ compound remains Heisenberg-like at all
temperatures, with its transition to the N\'{e}el state due to the interlayer
interactions. Effects of field-induced anisotropy have been observed.
