Well, if your RNA is contaminated with genomic DNA, then adding more RNA will simply result in you adding more DNA too, putting you right back to square one. So don't do that.

It might be worth running your RNA itself on a gel to see if it's really full of genomic DNA: mix a little bit of your RNA with DNA-gel loading buffer, plus formamide to about 60-80%, heat it to 65 for five mins, crash cool on ice, run on a normal DNA gel.

Good quality RNA should give you two sharp ribosomal bands, with a faint background smear corresponding to all the different mRNAs. Genomic DNA will usually be sufficiently long and unwieldy to barely enter the gel at all (so a fairly messy band right up by the well is usually what it looks like). 

If you think you have a lot of genomic DNA, you could DNAse treat your RNA, then heat inactivate the DNAse and reprecipitate your RNA. I don't generally like doing this because the thought of adding nucleases, even DNA-specific ones, to my precious RNA makes me shiver, but I know many people do this (I just use intron spanning primers).

So there's that.

More pertinently, you should probably supply more detail regarding what you're seeing on the gel when you run your PCR. Usually if you're designing intron-spanning primers, you should be spanning a LARGE intron (like, 10,000 or more basepairs), and amplifying a cDNA region that's relatively small (100-500bp). If you do this, then you shouldn't see any product even with genomic contamination because your cycling conditions should never be generous enough to allow products that long to amplify. 

Even with a shorter intron, using primers that amplify a short cDNA region (or the corresponding slightly longer genomic DNA region) will favour the shorter product anyway, as it takes less time to synthesise, so is more likely to double each cycle.

A final point to consider is this: invariably, on a cell-by-cell basis, the number of transcripts of any given gene, IF EXPRESSED, will be equal to or greater than the number of genomes, because the number of genomes per cell will always be "one" (dividing cells notwithstanding), and the number of transcripts will either be zero (not expressed) or "one or more". 

This means that even in a situation with heavy genomic DNA contamination, if your gene is being expressed at all, it should ALWAYS give you some product of your expected size. If you're generating genomic DNA bands and nothing else, then it's probably safe to assume that your expression is zero, or very, very low.

(or your cDNA synthesis didn't work, or your RNA is degraded)

I agree with all of John's points.  Even highly purified DNAse will degrade RNA by divalent catalyzed hydrolysis.  Try collecting 50% of the aqueous phase. Be sure the TRIZOL is pH4.

Protocol for First Strand cDNA Synthesis

The following protocol is optimized to generate first-strand

cDNA for use in two-step RT-PCR.

Mix and briefly centrifuge all components after thawing,

keep on ice.

1. Add into sterile, nuclease-free tube on ice in the

indicated order:

Template

RNA

total RNA

or

poly(A) RNA

or

specific RNA

0.1 ng-5 µg

10 pg-500 ng

0.01 pg-0.5 µg

Primer

Oligo(dT)18 (#SO131)

or

Random hexamer (#SO142)

or

gene-specific primer

0.5 µg (100 pmol)

0.2 µg (100 pmol)

15-20 pmol

DEPC-treated water (#R0601) to 12.5 µl

2. Optional: If RNA template is GC rich or is known to

contain secondary structures, mix gently, centrifuge

briefly and incubate at 65°C for 5 min, chill on ice,

briefly centrifuge and place on ice.

3. Add the following components in the indicated order:

5X Reaction Buffer 4 µl

Thermo Scientific RiboLock RNase

Inhibitor (#EO0381) 0.5 µl (20 u)

dNTP Mix, 10 mM each

(#R0191)

2 µl

(1 mM final concentration)

RevertAid Reverse Transcriptase 1 µl (200 u)

Total volume 20 µl

Mix gently and centrifuge briefly.

4. If oligo(dT)18 primer or gene-specific primer is used,

incubate 60 min at 42°C.

If random hexamer primer is used, incubate 10 min at

25°C followed by 60 min at 42°C.

For transcription of GC rich RNA reaction temperature

can be increased to 45°C.

5. Terminate the reaction by heating at 70°C for 10 min.

Do not heat-inactivate enzyme prior to analysis of long

cDNA to avoid cleavage.

Hi, first treat your RNA with DNAse or use other kit (column based, QIAGEN for example).