Beignet is the Intel-developed solution for handling OpenCL support by their open-source Linux graphics stack. Intel isn't using Gallium3D and its Clover state tracker for doing OpenCL but have their own home-grown GPGPU solution. Beignet still isn't ready for end-user use, but it's slowly getting into shape.
My last update on Intel's Beignet OpenCL was back in December while now a commit landing today to help out the project is LLVM/Clang 3.5 support. It turns out LLVM/Clang 3.3 is buggy for Beignet and some bugs were fixed in LLVM/Clang 3.4 but there's still some bugs outstanding there, but they have been fixed for LLVM/Clang 3.5. LLVM 3.5 should be officially released around mid-2014.
Other recent interesting activity for this Intel Linux OpenCL implementation can be found via Beignet on Anzwix or by visiting the FreeDesktop.org page that also includes setup instructions. Beignet has near complete support for OpenCL 1.1 and besides needing LLVM it also depends upon Mesa Git with GBM enabled, libdrm, and various X components. The bring-up has mostly been focused on Ivy Bridge (the first generation with a CL-capable HD Graphics GPU).
In a bit better shape is still the Radeon OpenCL stack that also depends upon the latest kernel, Mesa Git, and LLVM for the best support. Hopefully by the end of 2014 we will see more appropriate "out of the box" OpenCL support by Linux graphics drivers as for now I can only recommend the binary AMD and NVIDIA graphics drivers on Linux for a good and reliable OpenCL/GPGPU experience.