How about this:

use strict; use warnings;

my $letters = <DATA>;
my @letters = $letters =~ /\w+/g;

my %out;
while(<DATA>) {
  next if /SampleID/;
  chomp;
  my @recs = split /\s+/;
  my $ctr = 2;
  for my $letter (@letters) {
    $out{$letter} .= "$letter\t".(join "\t", @recs[0,1,$ctr++])."\n";
  }
}

for my $letter (@letters) {
  print $out{$letter};
}

__DATA__
               A     C     F
SampleID        Time    ObsConc ObsConc ObsConc
5        24      2.27E+06        687.02  32521.94
5        168     1.92E+06        525.02  22198.44
5        12      2.94E+06        896.39  41331.61
5        -0.5    2.23E+06        942.34  40616.49
5        8       4.03E+06        1371.32 45863.69
6        24      1.02E+06        1057.89 46341.04
6        168     3.14E+06        4987.32 42166.08
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This assumes that the line with A C F is potentially variable and due to change. Otherwise it could be further simplified by hardcoding it. It also assumes that you always repeat the first two columns.

It builds the output fully in memory, which might be a problem if you have HUGE input. In order to minimize memory consumption, one could use temporary files for each of the letters and merge them later.

UPDATE: No statement about the "ugliness" of this or other code is implied...