You are on the right track, and to get values for a particular time period, you have to crossjoin using the current context. Also, use DistinctCount instead of Count...
Here is a sample query from AdventureWorks:
WITH member [Measures].[Product Count] AS
DistinctCount([Date].[Fiscal Weeks].CurrentMember * [Product].[Product Categories].[Product].Members * {[Measures].[Internet Order Quantity]})
SELECT {[Measures].[Product Count]} ON COLUMNS,
[Date].[Fiscal Weeks].[Fiscal Week] ON ROWS
FROM [Adventure Works]