This just came to me as one possible solution. I have no idea as to performance, but I thought it would be an interesting way to solve the problem. I tested that it works in a simple situation (I didn't code to account for NULLs). Feel free to give it a test to see if it performs well for you.
The table that I used included an id (my_id). That could really be any column that is unique within the group (grp_id), so it could be a date column or whatever.
;WITH CTE AS (
SELECT
T1.my_id,
T1.grp_id,
CAST(T1.my_str AS VARCHAR) AS my_str
FROM
dbo.Test_Group_Concat T1
WHERE NOT EXISTS (SELECT * FROM dbo.Test_Group_Concat T2 WHERE T2.grp_id = T1.grp_id AND T2.my_id < T1.my_id)
UNION ALL
SELECT
T3.my_id,
T3.grp_id,
CAST(CTE.my_str + T3.my_str AS VARCHAR)
FROM
CTE
INNER JOIN dbo.Test_Group_Concat T3 ON
T3.grp_id = CTE.grp_id AND
T3.my_id > CTE.my_id
WHERE
NOT EXISTS (SELECT * FROM dbo.Test_Group_Concat T4 WHERE
T4.grp_id = CTE.grp_id AND
T4.my_id > CTE.my_id AND
T4.my_id < T3.my_id)
)
SELECT
CTE.grp_id,
CTE.my_str
FROM
CTE
INNER JOIN (SELECT grp_id, MAX(my_id) AS my_id FROM CTE GROUP BY grp_id) SQ ON
SQ.grp_id = CTE.grp_id AND
SQ.my_id = CTE.my_id
ORDER BY
CTE.grp_id