It seems that if you provide a TestSettings file for the Solution then you can uncheck the "Enable deployment" option and stop mstest from trying to run from the ...TestResults\...\out
folder where it doesn't copy your extra files (unless you make them a deployment option).
This is also useful if you depend on the extra files being in a preserved folder structure because Deployment items all seem to be copied directly (flat) into the temporary run folder (out) if you use the Deployment, Add Folder option in the TestSettings (answers above suggest you can keep the structure if you add each item as its own DeploymentItem).
For me it worked fine running tests directly in Visual Studio (i.e. my extra files in their structure were found and used by tests) because I had created a TestSettings file for another reason long ago (which has Enable deployment unchecked), but not when TeamCity ran mstest to run tests because I hadn't specified that the TestSettings file should be used.
To create a TestSettings file in Visual Studio, right click on the Solution and choose New Item, and select the TestSettings template. To use the TestSettings file at the command prompt of mstest.exe add the option, /testsettings:C:\Src\mySolution\myProject\local.testsettings
(or add as an extra command line option in TeamCity with appropriate path)