Improved understanding of heat transfer in thermoacoustic heat exchangers is desirable for the design of practical thermoacoustic devices. Such heat transfer is transverse (as opposed to axial) with respect to the stack pores. Current thermoacoustic analyses [e.g., Swift, J. Acoust. Soc. Am. 84, 1145--1180 (1988)] cannot treat time-average transverse heat transfer because they assume that the time-average temperature of the gas in any section is the same as that of the immediately adjacent section of the stack. The present analysis lifts this restriction by augmenting Swift's treatment of the oscillating temperature fields in gas and stack with an analogous treatment of the time-average temperature fields. Because the goal is to develop physical understanding, rather than to obtain quantitatively precise results, simplifying assumptions are introduced. The main assumption is that the gas and stack can be treated as essentially one-dimensional entities, within which any transverse temperature variations may be neglected. Within the approximations, the equations may be solved analytically. Results compare favorably with recent numerical work by Cao et al. [J. Acoust. Soc. Am. 99, 3456 (1996)]. Applications to thermoacoustic heat exchangers will be presented.