Acknowledge grants (which may indicate conflict of interest) in double blind review?
I think the best approach is, as recommended in the comments, to cite the funding organization but not the particular grant.  In fact, I have recently encountered some journals that request this information explicitly as part of the submission meta-data, which I think is very wise.

Citing only the funding organization may, indeed, be extremely vague information: for example, "US National Science Foundation" tells you pretty much nothing.  In general, however, this is already enough information to determine potential conflicts of interest:


If the funder is a broad-mission government agency, there there is generally no meaningful conflict of interest to consider.
When funding comes from a private organization, such as a company or a foundation, that information is generally enough to identify potential conflicts of interest.  It doesn't matter which division of an oil or pharmaceutical company is funding a project---all of their research projects will still be expected to relate to their bottom line somehow.