It seems to me that given the current schema you have, that you will need to retrieve the appropriate time IDs from the time dimension table which meet your search criteria, and then search for matching rows in the fact table. Depending on the granularity of your time dimension, you might want to check the performance of doing either (SQL Server examples):
A subselect:
SELECT X FROM FOO WHERE TIMEID IN (SELECT ID FROM DIMTIME WHERE HOUR >= DATEPART(HOUR, CURRENT_TIMESTAMP()) AND DATEID IN (SELECT ID FROM DIMDATE WHERE DATE = GETDATE())
An inner join:
SELECT X FROM FOO INNER JOIN DIMTIME ON TIMEID = DIMTIME.ID WHERE HOUR >= DATEPART(HOUR, CURRENT_TIMESTAMP()) INNER JOIN DIMDATE ON DATEID = DIMDATE.ID WHERE DATE = GETDATE()
Neither of these are truly attractive options.
Have you considered that you may be querying against a cube that is intended for roll-up analysis and not necessarily for "last X" analysis?
If this is not a "roll-up" cube, I would agree with the other posters in that you should re-stamp your fact tables with better keys, and if you do in fact intend to search off of hour frequently, you should probably include that in the fact table as well, as any other attempt will probably make the query non-sargable (see What makes a SQL statement sargable?).
Microsoft recommends at http://msdn.microsoft.com/en-us/library/aa902672%28v=sql.80%29.aspx that:
In contrast to surrogate keys used in other dimension tables, date and time dimension keys should be "smart." A suggested key for a date dimension is of the form "yyyymmdd". This format is easy for users to remember and incorporate into queries. It is also a recommended surrogate key format for fact tables that are partitioned into multiple tables by date.
Best luck!