In this paper we consider implementing optical burst switching as a technology for building Grids with computationally intensive requirements. This architecture has been referred to as Grid-over-OBS (GoOBS). First, we briefly describe the proposed layered Grid architecture and show how OBS can be positioned within the Grid architecture. Then, we present a generic framework for anycast routing in the context of GoOBS when requests don?t have an explicit destination address and they can be serviced by any appropriate Grid resource. We also develop several algorithms to support anycasting when only a single copy of a request is transmitted. Through simulation analysis, we show the performance of our anycast algorithms and compare them with traditional shortestpath unicast routing in which all jobs have specific addresses. Our performance analysis focuses on blocking probability of requests and average end-to-end delay.