AFAIK as of Exchange 2010, category is a multi value field so it does not work with search filters. However, you can search categories using AQS. The following code should do the trick.
ExchangeService service = GetService();
var iv = new ItemView(1000);
string aqs = "System.Category:Processed";
FindItemsResults<Item> searchResult = null;
do
{
searchResult = service.FindItems(WellKnownFolderName.Inbox, aqs, iv);
foreach (var item in searchResult.Items)
{
Console.WriteLine(item.Subject);
}
iv.Offset += searchResult.Items.Count;
} while (searchResult.MoreAvailable == true);