Associate SQL Server Queries with Databases Is there a way via T-SQL and DMVs to show which database a query was run against?  

The goal is to look at CPU and I/O by query by database so I can redistribute the databases as I reconfigure the environment.

I do not want to use trace, profiler or extended events if I can help it. I also do not want to shred XML if I can avoid it. I am looking for history not just active.
You have to concentrate on Memory, Disk IO and CPU utilization to balance out your environment.

Best is to start collecting and dumping the results in a utility database and to give you a good baseline using sql agent on a scheduled basis.

Alternatively, you can use Performance data collector (applies to sql server 2008 and up)



MEMORY:

You can look into buffer pool usage (what databases are occupying more space in buffer pool ?)

From :Performance issues from wasted buffer pool memory

SELECT
    (CASE WHEN ([database_id] = 32767)
        THEN N'Resource Database'
        ELSE DB_NAME ([database_id]) END) AS [DatabaseName],
    COUNT (*) * 8 / 1024 AS [MBUsed],
    SUM (CAST ([free_space_in_bytes] AS BIGINT)) / (1024 * 1024) AS [MBEmpty]
FROM sys.dm_os_buffer_descriptors
GROUP BY [database_id];
GO




DISK IO:

In terms of Disk IO, you need to move away the mdf/ldf files having the most disk IO to a better/faster LUN/Array :

From : How to examine IO subsystem latencies from within SQL Server

SELECT
    [ReadLatency] =
        CASE WHEN [num_of_reads] = 0
            THEN 0 ELSE ([io_stall_read_ms] / [num_of_reads]) END,
    [WriteLatency] =
        CASE WHEN [num_of_writes] = 0
            THEN 0 ELSE ([io_stall_write_ms] / [num_of_writes]) END,
    [Latency] =
        CASE WHEN ([num_of_reads] = 0 AND [num_of_writes] = 0)
            THEN 0 ELSE ([io_stall] / ([num_of_reads] + [num_of_writes])) END,
    [AvgBPerRead] =
        CASE WHEN [num_of_reads] = 0
            THEN 0 ELSE ([num_of_bytes_read] / [num_of_reads]) END,
    [AvgBPerWrite] =
        CASE WHEN [num_of_writes] = 0
            THEN 0 ELSE ([num_of_bytes_written] / [num_of_writes]) END,
    [AvgBPerTransfer] =
        CASE WHEN ([num_of_reads] = 0 AND [num_of_writes] = 0)
            THEN 0 ELSE
                (([num_of_bytes_read] + [num_of_bytes_written]) /
                ([num_of_reads] + [num_of_writes])) END,
    LEFT ([mf].[physical_name], 2) AS [Drive],
    DB_NAME ([vfs].[database_id]) AS [DB],
    [mf].[physical_name]
FROM
    sys.dm_io_virtual_file_stats (NULL,NULL) AS [vfs]
JOIN sys.master_files AS [mf]
    ON [vfs].[database_id] = [mf].[database_id]
    AND [vfs].[file_id] = [mf].[file_id]
-- WHERE [vfs].[file_id] = 2 -- log files
-- ORDER BY [Latency] DESC
-- ORDER BY [ReadLatency] DESC
ORDER BY [WriteLatency] DESC;
GO


Also refer to Leveraging sys.dm_io_virtual_file_stats



CPU:

From : Brent's answer 

SELECT total_worker_time/execution_count AS AvgCPU  
, total_worker_time AS TotalCPU
, total_elapsed_time/execution_count AS AvgDuration  
, total_elapsed_time AS TotalDuration  
, (total_logical_reads+total_physical_reads)/execution_count AS AvgReads 
, (total_logical_reads+total_physical_reads) AS TotalReads
, execution_count   
, SUBSTRING(st.TEXT, (qs.statement_start_offset/2)+1  
, ((CASE qs.statement_end_offset  WHEN -1 THEN datalength(st.TEXT)  
ELSE qs.statement_end_offset  
END - qs.statement_start_offset)/2) + 1) AS txt  
, query_plan
FROM sys.dm_exec_query_stats AS qs  
cross apply sys.dm_exec_sql_text(qs.sql_handle) AS st  
cross apply sys.dm_exec_query_plan (qs.plan_handle) AS qp 
ORDER BY 1 DESC


A gold mine of DMV related queries : SQL Server Diagnostic Information Queries for September 2014

For completeness, if you are using sql server 2012, you can use the system health reporting dashboard for Visualizing sp_server_diagnostics results.