I modified @Phil Sandler's code to remove the effect of calling GETDATE() (there may be hardware effects/interrupts involved??), and made rows the same length.

[There have been several articles since SQL Server 2000 relating to timing issues and high-resolution timers, so I wanted to minimise that effect.]

In simple recovery model with data and log file both sized way over what is required, here are the timings (in seconds): (Updated with new results based on exact code below)

       Identity(s)  Guid(s)
       ---------    -----
       2.876        4.060    
       2.570        4.116    
       2.513        3.786   
       2.517        4.173    
       2.410        3.610    
       2.566        3.726
       2.376        3.740
       2.333        3.833
       2.416        3.700
       2.413        3.603
       2.910        4.126
       2.403        3.973
       2.423        3.653
    -----------------------
Avg    2.650        3.857
StdDev 0.227        0.204

The code used:

SET NOCOUNT ON

CREATE TABLE TestGuid2 (Id UNIQUEIDENTIFIER NOT NULL DEFAULT NEWSEQUENTIALID() PRIMARY KEY,
SomeDate DATETIME, batchNumber BIGINT, FILLER CHAR(88))

CREATE TABLE TestInt (Id Int NOT NULL identity(1,1) PRIMARY KEY,
SomeDate DATETIME, batchNumber BIGINT, FILLER CHAR(100))

DECLARE @Numrows INT = 1000000

CREATE TABLE #temp (Id int NOT NULL Identity(1,1) PRIMARY KEY, rowNum int, adate datetime)

DECLARE @LocalCounter INT = 0

--put rows into temp table
WHILE (@LocalCounter < @NumRows)
BEGIN
    INSERT INTO #temp(rowNum, adate) VALUES (@LocalCounter, GETDATE())
    SET @LocalCounter += 1
END

--Do inserts using GUIDs
DECLARE @GUIDTimeStart DateTime = GETDATE()
INSERT INTO TestGuid2 (SomeDate, batchNumber) 
SELECT adate, rowNum FROM #temp
DECLARE @GUIDTimeEnd  DateTime = GETDATE()

--Do inserts using IDENTITY
DECLARE @IdTimeStart DateTime = GETDATE()
INSERT INTO TestInt (SomeDate, batchNumber) 
SELECT adate, rowNum FROM #temp
DECLARE @IdTimeEnd DateTime = GETDATE()

SELECT DATEDIFF(ms, @IdTimeStart, @IdTimeEnd) AS IdTime, DATEDIFF(ms, @GUIDTimeStart, @GUIDTimeEnd) AS GuidTime

DROP TABLE TestGuid2
DROP TABLE TestInt
DROP TABLE #temp
GO

After reading @Martin's investigation, I re-ran with the suggested TOP(@num) in both cases, i.e.

...
--Do inserts using GUIDs
DECLARE @num INT = 2147483647; 
DECLARE @GUIDTimeStart DATETIME = GETDATE(); 
INSERT INTO TestGuid2 (SomeDate, batchNumber) 
SELECT TOP(@num) adate, rowNum FROM #temp; 
DECLARE @GUIDTimeEnd DATETIME = GETDATE();

--Do inserts using IDENTITY
DECLARE @IdTimeStart DateTime = GETDATE()
INSERT INTO TestInt (SomeDate, batchNumber) 
SELECT TOP(@num) adate, rowNum FROM #temp;
DECLARE @IdTimeEnd DateTime = GETDATE()
...

and here are the timing results:

       Identity(s)  Guid(s)
       ---------    -----
       2.436        2.656
       2.940        2.716
       2.506        2.633
       2.380        2.643
       2.476        2.656
       2.846        2.670
       2.940        2.913
       2.453        2.653
       2.446        2.616
       2.986        2.683
       2.406        2.640
       2.460        2.650
       2.416        2.720

    -----------------------
Avg    2.426        2.688
StdDev 0.010        0.032

I wasn't able to get the actual execution plan, as the query never returned! It seems a bug is likely. (Running Microsoft SQL Server 2008 R2 (RTM) - 10.50.1600.1 (X64))

Since the seconds are from 0 (1970-01-01 00:00:00 UTC), you should look for every multiple of 60

SELECT * FROM mytable WHERE MOD(TimeStamp,60)=0;

or if TimeStamp is indexed, you can do

SELECT T.* FROM
(SELECT TimeStamp FROM mytable WHERE MOD(TimeStamp,60)=0) M
INNER JOIN mytable T USING (TimeStamp);

Give it a Try !!!

SUGGESTION #1

You should store the timestamp of the minute and index it

ALTER TABLE mytable ADD COLUMN MinuteTimeStamp AFTER TimeStamp;
UPDATE mytable SET MinuteTimeStamp = TimeStamp - MOD(TimeStamp,60);
ALTER TABLE mytable ADD INDEX MinuteTimeStamp_UniqueKey_ndx (MinuteTimeStamp,UniqueKey);

Then, you can do MIN aggregation on MinuteTimeStamp.

SELECT MinuteTimeStamp,MIN(UniqueKey) UniqueKey
FROM mytable GROUP BY MinuteTimeStamp;

and use it get those records

SELECT B.* FROM
(SELECT MinuteTimeStamp,MIN(UniqueKey) UniqueKey
FROM mytable GROUP BY MinuteTimeStamp) A
INNER JOIN mytable B USING (UniqueKey);

It was tactfully pointed out that triggers would degrade performance

Perhaps doing INSERTs like this may help

INSERT INTO mytable (UniqueKey,TimeStamp,MinuteTimeStamp) VALUES
(
    uniquevalue,
    UNIX_TIMESTAMP(NOW()),
    UNIX_TIMESTAMP(NOW() - INTERVAL SECOND(NOW()) SECOND)
);

SUGGESTION #2

Since you have over 1000 columns (Ugh), perhaps a table of those minute timestamps would be better.

CREATE TABLE MinuteKeys
(
    MinuteTimeStamp INT UNSIGNED NOT NULL,
    UniqueKey INT UNSIGNED NOT NULL,
    PRIMARY KEY (UniqueKey)
    KEY MinuteTimeStamp_UniqueKey_ndx (MinuteTimeStamp,UniqueKey)
) ENGINE=MyISAM;
ALTER TABLE MinuteKeys DISABLE KEYS;
INSERT INTO MinuteKeys SELECT TimeStamp - MOD(TimeStamp,60),UniqueKey FROM mytable;
ALTER TABLE MinuteKeys ENABLE KEYS;

Then, you could use that table for the aggregation

SELECT B.* FROM
(SELECT MinuteTimeStamp,MIN(UniqueKey) UniqueKey
FROM MinuteKeys GROUP BY MinuteTimeStamp) A
INNER JOIN mytable B USING (UniqueKey);

EPILOGUE

Other suggestions are possible but you should really consider normalization of the table

See my post Too many columns in MySQL as to why